Characterization of some finite simple groups by the set of orders of vanishing elements and order

  • S. Askary Imam-Ali Univ., Tehran, Iran
Keywords: Vanishing element, finite simple groups, Vanishing prime graph

Abstract

UDC 512.5

Характеризацiя деяких скiнченних простих груп множиною порядкiв зникаючих елементiв та порядку

Let $G$ be a finite group. We say that an element $g$ of $G$ is a vanishing element if there exists an irreducible complex character $X$ of $G$ such that $X(g) = 0$. Ghasemabadi, Iranmanesh, Mavadatpour (2015), present the following conjecture: Let $G$ be a finite group and $M$ a finite nonabelian simple group such that $Vo(G)=Vo(M)$ and $|G|=|M|$. Then $G \cong M $. We answer in affirmative this conjecture for $M = ^2 D_{r+1}(2)$, where $r = 2^n - 1 \geq 3$ and either $2^r+1$ or $2^{r+1}+1$ is a prime number and $M = ^2 D_{r}(3)$, where $r = 2^n + 1 \geq 5$ and either $(3^{r-1}+1)/2$ or $(3^{r}+1)/4$ is prime.

References

S. Asgary, N. Ahanjideh, Characterization of $P SL(3, q)$ by nse, Math. Rep., 19 (60), № 4, 425 – 438 (2017).

S. Asgary, N. Ahanjideh, nse characterization of some finite simple groups, Sci. Ann. Comput. Sci., 3 (2), 797 – 806 (2016).

G. Chen, Further reflections on Thompson’s conjecture, J. Algebra, 218, 276 – 285 (1999), https://doi.org/10.1006/jabr.1998.7839 DOI: https://doi.org/10.1006/jabr.1998.7839

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of finite groups, Clarendon Press, New York (1985).

G. Y. Chen, On Frobenius and 2-Frobenius group, J. Southwest China Normal Univ., 20, № 5, 485 – 487 (1995).

P. Crescenzo, A Diophantine equation which arises in the theory of finite groups, Adv. Math., 17, 25 – 29 (1975), https://doi.org/10.1016/0001-8708(75)90083-3 DOI: https://doi.org/10.1016/0001-8708(75)90083-3

S. Dolfi, E. Pacific, L. Sanus, P. Spiga, On the vanishing prime graph of finite groups, J. Lond. Math.Soc., II. Ser., 82, № 1, 167 – 183 (2010), https://doi.org/10.1112/jlms/jdq021 DOI: https://doi.org/10.1112/jlms/jdq021

S. Dolfi, E. Pacific, L. Sanus, P. Spiga, On the vanishing prime graph of solvable groups, J. Group Theory, 13, № 2, 189 – 206 (2010), https://doi.org/10.1515/JGT.2009.046 DOI: https://doi.org/10.1515/jgt.2009.046

M. F. Ghasemabadi, A. Iranmanesh, M. Ahanjideh, A new characterization of some families of finite simple groups, Rend. Semin. Mat. Univ. Padova, 137, 57 – 74 (2017), https://doi.org/10.4171/RSMUP/137-3 DOI: https://doi.org/10.4171/RSMUP/137-3

M. F. Ghasemabadi, A. Iranmanesh, F. Mavadatpur, A new characterization of some finite simple groups, Sib. Math. J., 56, № 1, 78 – 82 (2015), https://doi.org/10.1134/s0037446615010073 DOI: https://doi.org/10.1134/S0037446615010073

I. M. Isaacs, Character theory of finite groups, Pure and Appl. Math., vol. 69, Acad. Press, New York (1976).

G. James, M. Liebeck, Representations and characters of groups, Cambridge Math. Textbooks (1993).

H. Shi, G. Y. Chen, Relation between $B_p(3)$ and $C_p(3)$ with their order components where $p$ is an odd prime, J. Appl. Math. Inform., 27, 653 – 659 (2009).

A. V. Vasil’ev, E. P. Vdovin, An adjacency ceriterion for the prime graph of a finite simple group, Algebra and Logic, 44, 381 – 406 (2005), https://doi.org/10.1007/s10469-005-0037-5 DOI: https://doi.org/10.1007/s10469-005-0037-5

J. S. Williams, Prime graph components of finite groups, J. Algebra, 69, 487 – 513 (1981)б https://doi.org/10.1016/0021-8693(81)90218-0 DOI: https://doi.org/10.1016/0021-8693(81)90218-0

J. Zhang, Z. Li, C. Shao, Finite groups whose irreducible characters vanish only on elements of prime power order, Int. Electron. J. Algebra, 9, 114 – 123 (2011).

J. Zhang, C. Shao, Z. Shen, A new characterization of Suzuki, $s$ simple groups, J. Algebra and Appl, 16, 1 – 6 (2017), https://doi.org/10.1142/S0219498817502164 DOI: https://doi.org/10.1142/S0219498817502164

K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math. Phys., 3, № 1, 265 – 284 (1892), https://doi.org/10.1007/BF01692444 DOI: https://doi.org/10.1007/BF01692444

Published
23.11.2021
How to Cite
Askary, S. “Characterization of Some Finite Simple Groups by the Set of Orders of Vanishing Elements and Order”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 11, Nov. 2021, pp. 1443 -50, doi:10.37863/umzh.v73i11.1069.
Section
Research articles